Background

First consider the alpha particle model. The alpha particle model of nuclei is the notion that the neutrons and protons in a nucleus, whenever possible, are arranged in alpha particles. This model has been around since the 1930's, never verified but never entirely discredited. A couple of factors in its favor are the emission of alpha particles
from nuclei and that it would explain most of the binding energy of nuclei.
The binding energy of a deuteron (one neutron and one proton) is only about 2.2 million electron volts (MeV). The binding energies of the triteron (two neutrons and one proton) and the helium 3 nuclide are about 8.5 and 7.7 MeV, respectively.
Then there is the binding energy of 28.3 MeV for the alpha particle made up of
two neutrons and two protons. The binding energies of nuclides larger than an
alpha particle seem to include the binding energy of alpha particles.

The Forces between Nucleons

Two nucleons can form a spin pair which involves a binding energy to 2 to 3
MeV. A neutron can form a spin pair with another neutron and alsowith a proton, but
no more than one of each type. Likewise a proton can form a spin pair with another
proton and with a neutron. This leads to chains of protons and neutrons. The
significance of such chains is dealt with later.

There is another force between nucleons which is called the strong force.
The strong force is not exclusive. One nucleon may interact with all of the other
nucleons in a nucleus, but the magnitude of an interaction is dependent upon the
distance separating the nucleons.

The Excess Binding Energies of the Alpha Nuclides

The excess binding energy of a nuclide is its binding energy less the binding
energy that would arise from the formation of alpha particles from its nucleons.
There are 25 nuclides that could contain an integral number of alpha particles.
Their excess binding energies are shown in the graph below.

The bent line appearance of the graph is evidence of some sort of shell structure. However it is not merely a matter of three shells. The incremental
change in excess binding energy gives the details of the shell structures.

The black numbers are the possible number of alpha particles; the red numbers are
the number of neutrons (and also the number of protons). The numbers at which the excess
binding energy shows an abrupt drop are just the conventional magic numbers
augmented with 14, a number shown elsewhere to also be a magic number. The other
additional magic number is 6, which in the graph shows up to have an incremental
excess binding energy comparable to that of 8. The sharp drops in excess
binding energy occur when a shell is filled and the next alpha particle must go into a higher shell. These figures may hold the key to the spatial structure
of nuclides, but more on this later.

The Alternate to the Alpha Particle Model of the Structure of Nuclei

A neutron can form a spin pair with another neutron and with a proton. The same
applies for a proton. This means that chains of nucleons can be formed involving
a neutron pair being linked to a proton pair which in turn is linked to another
neutron pair. Thus a chain contain sections of the form -n-p-p-n- or equivalently -p-n-n-p-.
Such an arrangement is depicted below with the red dots representing protons and the black ones neutrons. The lines between the dots
represent spin pair bonds. The ring rotates to keep the neutron-proton paisr from collapsing under the simultaneous attraction through the
spin pair formation and the strong force. The neutron-neutron and proton-proton pairs are stabilized through a balancing of their attraction
through the attraction through spin pair formation and the repulsion through the nuclear strong force.

This is not an exact description of the spatial arrangement of the nucleons in
such a chain. The depiction of an alpha particle in style of the above would be
the figure shown on the left below, whereas a more proper representation would be
the tetrahedral arrangement shown on the right.

Here is a better visual depiction of an alpha particle.

The chains may be closed forming a ring.

Such chains are made up of modules involving two neutrons and two protons.
In a module each neutron is involved in two spin pairs and three interactions
and likewise for each proton
just as in an alpha particle. Here is what is meant by the term alpha module.

Thus the potential and kinetic energies and binding
energy is the same as in an alpha particle. An alpha particle is, in effect, a
chain of length one
alpha module.

Nuclear Shells

The concept of nuclear shells goes back to Maria Goeppert-Mayer and Hans Jensen.
They found that there are certain numbers of nucleons that are associated with
nuclear stability. They called these numbers magic numbers. Their magic
numbers were {2, 8, 20, 28, 50, 82, 126}. These numbers correspond to filled shells. The same numbers apply to neutrons and protons. In other studies it was found that 6 and 14 are also magic numbers.
There is a simple algorithm that generates the sequence {2, 6, 14, 28, 50, 82, 126}. This suggest that the sequence corresponds to the main sequence of
filled shells {2, 6, 14, 28, 50, 82, 126} and {8, 20} correspond to filled subshells within shells. For the sequence {2, 6, 14, 28, 50, 82, 126} the capacities of the shells are {2, 4, 8, 14, 22, 32, 44}.

Thus from the nuclear shell model it is known that the first neutron shell contains
two neutrons. Likewise the first proton shell contains two protons. The two
neutrons and two protons constitute an alpha particle so the first shell
is just an alpha particle. This says that the center of a nucleus is an alpha
particle. The existence of one and only one alpha particle in a nucleus explains why generally in
nuclear reaction there is only one alpha particle emitted.

The second neutron shell contains four neutrons and likewise for the second
proton shell. This is two alpha modules. The combination of the first and
second shells is the Carbon 12 nuclide, a very stable nuclide. In contrast
a nuclide with four neutrons and four protons is the Beryllium 8 nuclide that
has a half life of less than 4 seconds. It comes apart
into two alpha particles.

The nuclide with four alpha modules is Oxygen 16. This might be a ring of two
modules and two alpha particles or, perhaps, two rings of two modules each.

The incremental excess binding energies (IXBE) of the third and fourth alpha modules are
both about 8 MeV. The IXBE for the fifth module drops to about 5.5 MeV, but the
IXBE for the sixth jumps to about 10 MeV, making the average for the fifth and
sixth module about the same as that of the third and fourth. The IXBE for the
seventh module rises a bit higher than that of the sixth, suggesting that the
seventh module completes some special arrangement. The nuclide with seven alpha
modules is Silicon 28.

The IXBE for the eighth module drops down to about the same level as for the
third and fourth modules and the IXBE of the ninth and tenth are about the
same as that of the eighth.

The IXBE for the eleventh module drops sharply and to about the level of the
fifth module. For the twelfth, thirteenth and fourteenth modules the IXBE rises to
a new high level at about 8.5 MeV. The fourteenth module completes a shell and
thereafter the IXBE drops to levels between 3 and 4 MeV. In this fifth shell
the IXBE rises to a peak of about 4 MeV for the twentieth module and then
falls and then rises to another peak of about 4 MeV for the 25th module.

The 20th module is in the exact middle for the fifth shell. Likewise the
low points at the fifth and eleventh modules are exactly in the middle of
their shells.

The fifth shell consists of eleven modules. The pattern of rising to a peak
at the sixth module in the shell suggests that a subshell of six is being filled.
The remaining five places in the shell are then filled, with the 25th module
completing the shell.

It is notable in the above graph that the points that unusual in level compared with other nearby points are exactly in the middle. For eample, the value for
20 alphas (40 neutrons and 40 protons) five cases to the left and five to the
right. Likewise the value for 5 alphas (ten neutron and ten protons) has two cases
to the left and two to the right. Furthermore the case for 11 alphas has
three cases to the left and three to the right.